Related papers: Remarks on aleph_1-metrizable not metrizable first…
We study the question of when an uncountable ccc topological space $X$ contains a ccc subspace of size $\aleph_1$. We show that it does if $X$ is compact Hausdorff and more generally if $X$ is Hausdorff with $\mathrm{pct}(X) \leq \aleph_1$.…
In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric spaces are Hausdorff and also first countable. Moreover, we assert that every separable $\mathcal{F}$-metric space is second countable. Additionally, we acquire…
A cuf space (set, resp.) is a space (set, resp.) which is a countable union of finite subspaces (subsets, resp.). It is proved in $\mathbf{ZF}$ (with the absence of the axiom of choice) that all countable unions of cuf (denumerable, resp.)…
We show that square(theta) implies that there is a first countable <theta-collectionwise Hausdorff space that is not weakly theta-collectionwise Hausdorff. We also show that in the model obtained by Levy collapsing a weakly compact…
We show that all sufficiently nice $\lambda$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $\kappa \le \mathfrak{b}$ there is a countable dense homogeneous metric space…
It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says…
In this paper, some features of countably $\alpha$-compact topological spaces are presented and proven. The connection between countably $\alpha$% -compact, Tychonoff, and $\alpha$-Hausdorff spaces is explained. The space is countably…
This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof…
We answer several questions of V. Tka\v{c}uk from [Point-countable $\pi$-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55--69.] by showing that (1) there is a ZFC example of a first countable, 0-dimensional…
In this note we prove several theorems that are related to some results and problems from [6]. We answer two of the main problems that were raised in [6]. First we give a ZFC example of a Hausdorff space in $C(\omega_1)$ that has…
This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators…
In this paper I will construct a non-separable hereditarily Lindelof space (L space) without any additional axiomatic assumptions. I will also show that there is a function f from [omega_1]^2 to omega_1 such that if A,B, subsets of omega_1,…
A dual pair formulation for asymmetric locally convex spaces is developed that strictly generalises the ordinary vector space setting. The concept of a polar topology carries over to the asymmetric case and some familiar results are…
As defined in [1], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite} closed subsets of it intersect. Our main result answers the two main questions of [1] by…
We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging omega-sequence or a non-trivial converging omega_1-sequence. We establish that this dichotomy holds in a variety of models;…
Assuming the Continuum Hypothesis, there is a compact first countable connected space of weight aleph_1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add…
In this paper a construction of a metrizable zero-dimensional CDH space $X$ such that $X^2$ has exactly $\mathfrak{c}$ countable dense subsets is provided. Furthermore, it is shown that the space can be constructed consistently co-analytic.…
Tkachuk and Wilson proved that a regular first countable cellular-compact space has cardinality not exceeding the continuum. In the same paper they asked if this result continues to hold for Hausdorff spaces. Xuan and Song considered the…
In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among other…
A. V. Arhangel'ski\u{i} introduced in 2012, when he was visiting the department of Mathematics at King Abduaziz University, new weaker versions of normality, called \it $C$-normality, \rm and \it countable normality. \rm The purpose of this…